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March 2014 On comparison of clustering properties of point processes
Bartłomiej Błaszczyszyn, D. Yogeshwaran
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Adv. in Appl. Probab. 46(1): 1-20 (March 2014). DOI: 10.1239/aap/1396360100

Abstract

In this paper, we propose a new comparison tool for spatial homogeneity of point processes, based on the joint examination of void probabilities and factorial moment measures. We prove that determinantal and permanental processes, as well as, more generally, negatively and positively associated point processes are comparable in this sense to the Poisson point process of the same mean measure. We provide some motivating results on percolation and coverage processes, and preview further ones on other stochastic geometric models, such as minimal spanning forests, Lilypond growth models, and random simplicial complexes, showing that the new tool is relevant for a systemic approach to the study of macroscopic properties of non-Poisson point processes. This new comparison is also implied by the directionally convex ordering of point processes, which has already been shown to be relevant to the comparison of the spatial homogeneity of point processes. For this latter ordering, using a notion of lattice perturbation, we provide a large monotone spectrum of comparable point processes, ranging from periodic grids to Cox processes, and encompassing Poisson point processes as well. They are intended to serve as a platform for further theoretical and numerical studies of clustering, as well as simple models of random point patterns to be used in applications where neither complete regularity nor the total independence property are realistic assumptions.

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Bartłomiej Błaszczyszyn. D. Yogeshwaran. "On comparison of clustering properties of point processes." Adv. in Appl. Probab. 46 (1) 1 - 20, March 2014. https://doi.org/10.1239/aap/1396360100

Information

Published: March 2014
First available in Project Euclid: 1 April 2014

zbMATH: 1295.60059
MathSciNet: MR3189045
Digital Object Identifier: 10.1239/aap/1396360100

Subjects:
Primary: 60E15 , 60G55
Secondary: 60D05 , 60G60

Keywords: association , clustering , determinantal point process , directionally convex ordering , permanental point process , perturbed lattice , point process , sub-Poisson point process , super-Poisson point process

Rights: Copyright © 2014 Applied Probability Trust

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Vol.46 • No. 1 • March 2014
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